We develop the foundation of the \textit{complex symplectic geometry} of Lagrangian subvarieties in a hyperk\"{a}hler manifold. We establish a characterization, a Chern number inequality, topological and geometrical properties of Lagrangian submanifolds. We discuss a category of Lagrangian subvarieties and its relationship with the theory of Lagrangian intersection.\newline \mbox{\phantom{aa}} We also introduce and study extensively a normalized \textit{Legendre transformation} of Lagrangian subvarieties under a birational transformation of projective hyperk\"ahler manifolds. We give a \textit{Pl\"{u}cker type formula} for Lagrangian intersections under this transformation.